無人機外文翻譯-四旋翼無人機位置和姿態(tài)跟蹤控制【中文4860字】【PDF+中文WORD】,中文4860字,PDF+中文WORD,無人機,外文,翻譯,四旋翼,位置,姿態(tài),跟蹤,控制,中文,4860,PDF,WORD 【中文4860字】 四旋翼無人機位置和姿態(tài)跟蹤控制 摘要: 一個綜合控制方法是提出要執(zhí)行的位置和姿態(tài)跟蹤小型四旋翼的動力學(xué)模型無人機（UAV），那里的動力學(xué)模型是欠驅(qū)動控制，高度耦合非線性的。首先，動力學(xué)模型分為全面啟動子系統(tǒng)和欠驅(qū)動子系統(tǒng)；其次，全面啟動子系統(tǒng)的控制器通過一種新的強大的終端滑?？刂疲ㄅ_積電）的算法，這是用來保證所有狀態(tài)變量在短時間內(nèi)收斂到自己想要的值，收斂時間是如此之小，狀態(tài)變量擔(dān)任時間不變量的欠驅(qū)動子系統(tǒng)，另外，在欠驅(qū)動子系統(tǒng)的控制器通過滑模控制（SMC）設(shè)計。此外，該子系統(tǒng)的穩(wěn)定性都證明了Lyapunov理論；最后，為了證明所提出的控制方法的魯棒性，空氣動力學(xué)的力和力矩，并作為外部擾動空氣阻力考慮在內(nèi)，得到的仿真結(jié)果表明，合成控制方法的立場和態(tài)度方面都有不錯的表現(xiàn)當(dāng)遇到外部干擾跟蹤。 關(guān)鍵詞：四旋翼無人機，欠驅(qū)動，新穎的魯棒臺積電，SMC，綜合控制 1.介紹 四旋翼無人飛行器（UAV）正被用于一些典型的任務(wù)，如搜索和救援任務(wù)，監(jiān)督，檢查，測繪，航空攝影和法律的強制執(zhí)行。 考慮到旋翼的動力學(xué)模型是一個欠驅(qū)動，高度耦合的和非線性的系統(tǒng)，很多控制策略，已經(jīng)開發(fā)了一類相似的系統(tǒng)。其中，滑模控制，這已引起研究人員的 矚目，一直是一個有用的和有效的控制算法，處理系統(tǒng)具有較大不確定性，隨時間變化的特性，非線性和有界外部干擾。該方法是基于定義指數(shù)穩(wěn)定的滑動面 作為機能缺失跟蹤誤差sandusing李亞普諾夫理論的 ，保證所有的狀態(tài)軌跡在有限時間到達這些表面，另外，這些表面是漸近穩(wěn)定，狀態(tài)軌跡滑動沿著這些表面，直到他們到達原點。但是，為了獲得快速跟蹤誤差收斂，期望的極點必須遠離原點選擇上的左半部分s平面，同時，這將反過來增加了控制器的增益，這是不可取的考慮，在實際系統(tǒng)中的致動器飽和。 與非取代了傳統(tǒng)的線性滑動面線性終端滑動面，更快的跟蹤誤差收斂是獲得通過終端滑模控制，終端的滑動模式已被證明是有效的，用于提供更快收斂比圍繞平衡點的線性超平面型滑模。臺積電提出了不確定動態(tài)系統(tǒng)與純料回分鐘。一個魯棒自適應(yīng)臺積電技術(shù)被用于正剛性連接的機械手具有不確定動態(tài)發(fā)展。一個全球性的非奇異臺積電剛性機械臂正在呈現(xiàn)。機器人系統(tǒng)的有限時間控制是通過兩個狀態(tài)反饋和動態(tài)輸出反饋控制研究。使用終端的滑動模式的一種新形式的剛性機械手的連續(xù)有限時間控制方案被建議。為了實現(xiàn)有限時間跟蹤控制中的轉(zhuǎn)子位置的非線性推力主動磁軸承系統(tǒng)的軸向，強勁的非奇異臺積電被賦予。然而，傳統(tǒng)的臺積方法不是最好的收斂時間，主要的原因是非線性滑模的收斂速度比時的狀態(tài)變量是接近平衡點的線性滑動模式慢。使用增強功能的滑動一個新的計劃，臺積電開發(fā)超平面對跟蹤誤差收斂到零的有限時間，提出了不確定性的單輸入和單輸出（SISO）非線性系統(tǒng)具有未知外部干擾的保證。在大多數(shù)現(xiàn)有的研究成果，在不確定的外部干擾都沒有考慮這些非線性系統(tǒng)。為了進一步展示的新穎TSMC的魯棒性，外部干擾被認為是進入非線性系統(tǒng)和被施加到所述控制器的設(shè)計。 圖1 四旋翼無人機 在這項工作中，我們結(jié)合兩部分組成控制，對于高精度的新穎的魯棒臺積電組件在完全致動子系統(tǒng)的跟蹤性能以及一個SMC 組件處理在欠驅(qū)動子系統(tǒng)的外部干擾。 盡管許多經(jīng)典，高階和SMC擴展策略，已經(jīng)開發(fā)了飛行控制器設(shè)計的四旋翼無人機，在報紙上這些策略被用來決定一個必要補償外部干擾，此外，其他的控制方法，如比例-積分-微分（PID）控制，回步平控制，開關(guān)模型預(yù)測姿態(tài)控制等等。已經(jīng)提出了用于在飛行控制器的設(shè)計，上述控制策略，已經(jīng)提出了為使旋翼穩(wěn)定在有限時間和空氣工藝的穩(wěn)定時間可能太長，以反映他們的表現(xiàn)，穩(wěn)定時間為四旋翼無人機，快速從一些意想不到的干擾中恢復(fù)至關(guān)重要的意義。為了減少時間，基于新穎的魯棒TSMC和SMC算法合成控制方法被應(yīng)用到的動態(tài)模型四旋翼無人機。合成控制方法，提出以保證所有的系統(tǒng)狀態(tài)變量在短時間內(nèi)收斂到他們的期望值。此外，狀態(tài)變量的收斂時間進行了預(yù)測通過由新穎的魯棒臺積電得出的方程式，這表現(xiàn)在以下幾個部分。 這項工作的組織安排如下：第2節(jié)提出了一個小的四旋翼無人機的動力學(xué)模型。合成控制方法是在第3節(jié)詳細的介紹。在第4節(jié)，仿真結(jié)果分析，以突出整體有效性和所設(shè)計的控制器的有效性。第5節(jié)的討論，這是基于不同的合成控制方案，提出了強調(diào)表現(xiàn)在這項工作中提出的綜合控制方法，其次是結(jié)束語在第6部分。 2. 旋翼模型 為了描述的旋翼模型的運動情況，顯然，位置坐標(biāo)是選擇。旋翼是建立在這一工作由主體框架B和接地E型如圖呈現(xiàn)。讓矢量表示旋翼的重心的位置和向量表示其在地球幀的線速度。向量表示旋翼的角速度在主體框架，表示的總質(zhì)量。表示重力加速度。表示從每個轉(zhuǎn)子的中心至重心的距離。 在旋翼的方向是由旋轉(zhuǎn)矩陣R給定：,其中R取決于三個歐拉角，這代表了翻滾，俯仰。且，，。 從變換矩陣到被給出 (1) 在旋翼的動力學(xué)模型可以由以下方程來描述 (2) 式中，Ki表示阻力系數(shù)和正的常數(shù)，，靜置螺旋槳的角速度，，，代表旋翼的轉(zhuǎn)動慣量，表示螺旋槳的轉(zhuǎn)動慣量，表示總瑟斯頓體在軸；和表示的側(cè)傾和俯仰的輸入；表示偏航力矩。，，，。其中表示由四個轉(zhuǎn)子所產(chǎn)生的推力和被認為是真正的控制輸入到動力系統(tǒng)，表示升力系數(shù);表示的力，力矩的比例因子。 3.綜合控制 與無刷電機相比，螺旋槳是很輕的，我們忽略的轉(zhuǎn)動慣量所引起的螺旋槳。式（2）是 分為兩部分： (3) (4) 其中公式（3）表示完全致動子系統(tǒng)（FAS），式（4）表示的欠驅(qū)動子系統(tǒng)（UAS）。對于FAS，一個新穎的魯棒TSMC用于保證其狀態(tài)變量在短時間內(nèi)收斂到其所需的值，然后，狀態(tài)變量被視為時間不變性，因此，UAS得到簡化。對于UAS，滑?？刂品椒ɡ?。特別合成控制方案在以下幾節(jié)介紹。 3.1一種新型強大的臺積電FAS 考慮到一個剛體旋翼的對稱性，然而，我們得到，和完全觸動子系統(tǒng)寫的是 (5) 為了開發(fā)跟蹤控制，滑動歧管被定義為 (6) 當(dāng)，，和是狀態(tài)變量的期望值。此外，該系數(shù)是正的，是正奇數(shù)整數(shù)讓和收斂時間的計算方法如下 (7) 根據(jù)公式（5）與S2和S4的時間導(dǎo)數(shù)，我們有 (8) 該控制器被設(shè)計 (9) 這里是積極的，也是正奇數(shù)整數(shù)且，根據(jù)控制器的狀態(tài)軌跡到達的區(qū)域滑動表面，在有限時間內(nèi)，時間被定義為 (10) 在 證明1為了說明該子系統(tǒng)是穩(wěn)定的，在這里，我們只選擇了狀態(tài)變量，和 Lyapunov得以理論應(yīng)用。 考慮到Lyapunov函數(shù) 調(diào)用方程（8）和（9）V1的時間導(dǎo)數(shù)導(dǎo)出 考慮到為正偶數(shù)而且。該子系統(tǒng)的狀態(tài)軌跡在有限時間收斂到期望的平衡點，因此，子系統(tǒng)是漸近穩(wěn)定的。 3.2 SMC的方式為無人機 在本節(jié)中，左右推拉的一類欠驅(qū)動系統(tǒng)的模式控器的細節(jié)被發(fā)現(xiàn)在SMC方法的無人機系統(tǒng)。 ，， 在欠驅(qū)動子系統(tǒng)是寫在一個級聯(lián)的形式 (11) 根據(jù)公式（9），我們可以選擇合適的參數(shù)，以保證控制律和偏航角ψ收斂到期望的/參考值在很短的時間。時，不隨時間變化，然后，是不隨時間變化和非奇異的矩陣，作為其結(jié)果是總推力和非零克服重力。 確定跟蹤誤差方程 (12) 其中所述載體來表示所希望的值的矢量 滑動歧管被設(shè)計成 (13) 其中常數(shù) 由于 ， 可以得到 (14) 圖2 位置，PID控制和SMC 圖3 坐標(biāo)，PID控制和SMC 證明2該子系統(tǒng)的穩(wěn)定性由李雅普諾夫說明理論如下： 考慮Lyapunov函數(shù) 調(diào)用（13）和（14），V的時間導(dǎo)數(shù)是 因此，在控制器，子系統(tǒng)的狀態(tài)軌跡可以達到，此后，在有限時間保持。 圖4 控制器，PID控制和SMC 表1 旋翼模型參數(shù) 表2 控制器參數(shù) 4.仿真結(jié)果與分析 在本節(jié)中，式中的四旋翼無人機的動力學(xué)模型，當(dāng)遇到外部干擾，式（2）用于測試所提出的合成控制方案的有效性和效率。典型的位置和姿態(tài)跟蹤的仿真在Matlab7.1.0.246/ Simulink中進行的，其配備了包括DUO E72002.53 GHz的CPU與2GB的內(nèi)存和100GB的固態(tài)硬盤驅(qū)動器的計算機。此外，該合成控制的性能通過被證實的與控制方法相比，它使用一個速率控制方法相比，有界的PID控制器和滑?？刂破鞯耐耆?qū)動子系統(tǒng),一個SMC方法的欠驅(qū)動子系統(tǒng)。 4.1 PID控制和SMC 在本節(jié)中，將PID控制和SMC方法的更多細節(jié)對于aquadrotor無人機已經(jīng)出臺,同時，仿真結(jié)果和分析，從而驗證的有效性綜合控制方案，可以發(fā)現(xiàn) 圖5 坐標(biāo)（X，Y，Z），新穎的魯棒臺積電和SMC 圖6 角，新穎的魯棒臺積電和SMC 模擬測試顯示在圖2-4，然而，研究科目略有改變，使具有明顯的比較以下模擬測試。 4.2 新穎的魯棒臺積電和SMC 在本節(jié)中，為了證明所提出的合成控制方法的有效性，已經(jīng)進行了四旋翼的位置和姿態(tài)跟蹤。 在旋翼的模擬測試的初始位置和角度的值是[0,0,0]和[0,0,0]。此外，該旋翼模型變量列于表1中。所需/參考位置和角度值在模擬測試中使用，此外，該控制器參數(shù)列于表2，仿真結(jié)果示于圖5-10。 整體控制方案管理，以有效地保持在有限時間的四旋翼水平位置和姿態(tài)，圖5和圖6有所展示，狀態(tài)變量z和ψ的有限時間收斂顯然比其他狀態(tài)變量更快，因此，它是安全的考慮俯仰角ψ為不隨時間變化1.163s之后。此外，高度Z達到1.779s后，控制器U1根據(jù)其參考值，因此，它是可靠的后1.779秒到考慮控制器U1為不隨時間變化。這些驗證矩陣Q是時間不變的短有限時間。在其他變量后約5秒達到他們的期望值。即使?fàn)顟B(tài)變量φ和θ的光滑曲線表明，它們有一定的振蕩幅度，該幅度是從-0.05弧度到0.05弧度不同。根據(jù)初始條件，參數(shù)和希望/參考值，狀態(tài)變量z和ψ的收斂時間是通過調(diào)用方程計算值基本一致。這表明所提出的合成控制方案的有效性。 圖7 線速度（U，V，W），新穎的魯棒臺積電和SMC 線速度和角速度，顯示在圖7和8，分別表現(xiàn)出相同的行為，相應(yīng)的位置和角度，事實上，這些狀態(tài)變量被驅(qū)趕到它們的穩(wěn)定狀態(tài)如預(yù)期。這再次證明了綜合控制方案的有效性。 滑動變量（S2，S4和s），示于圖9，如下的期望，因為所有的變量收斂到其滑動面。此外，如需要，為S2和S4的有限時間收斂明顯大于s的有限時間收斂速度更快。同樣的，這表現(xiàn)出相同的行為，在圖中所示5和6。 由圖可見，如圖10所示，可以發(fā)現(xiàn)，該四個控制輸入變量幾秒鐘后，分別收斂于穩(wěn)態(tài)值。此外，這也驗證了矩陣是不隨時間變化在短期有限的時間，盡管有較高的初始值，和幾乎沒有振動的振幅。這也表示其中趨勢的時間導(dǎo)數(shù)為零。因此， 在方程的方程組進行比較，（11）被大大簡化。 最后，提出了在所有的控制方法的魯棒性證明通過考慮氣動力和力矩，并作為外部干擾到旋翼的動力學(xué)模型空氣阻力。而且，這些干擾術(shù)語也適用于在控制器的設(shè)計。其結(jié)果是，這些擾動方面的影響是不可見的所有狀態(tài)變量，滑動變量，和控制器。 5.討論 廣泛的模擬測試已經(jīng)完成，評估不同的合成控制計劃，是基于四旋翼無人機的位置和姿態(tài)的跟蹤，它可以beclearly看出，雖然根據(jù)需要在有限時間所有的狀態(tài)變量收斂到他們的參考值時，收斂時間顯然是不同的。結(jié)果表明，基于該新穎的魯棒TSMC和SMC的合成控制方法是一種更可靠和更有效的方法來執(zhí)行跟蹤控制的旋翼UAV。 6.結(jié)論 這項工作研究的立場和態(tài)度使用基于上述控制算法所提出的控制方法跟蹤一個小四旋翼無人機的控制權(quán)，為了進一步測試設(shè)計的控制器的性能，隨著控制器的四旋翼的動力學(xué)模型進行仿真基于Matlab / Simulink的。主要結(jié)論概述如下。 （a）本六個自由度在有限時間分別收斂到其期望/參考值。 （b）該狀態(tài)變量z和ψ的收斂時間是與理論計算值（C）基本上是一致的相對于其他變量，俯仰角ψ和控制器成為時間變量在很短的時間（D）四個控制輸入變量收斂到在有限時間穩(wěn)定的價值觀， 和幾乎沒有振動振幅。所有上述情況，提出了合成控制方法的有效性和魯棒性已被證實，并且，所呈現(xiàn)的模擬結(jié)果是有希望的位置和姿態(tài)跟蹤控制的飛機。 致謝 本工作部分由中國國家自然科學(xué)基金科學(xué)基金（60905034）提供。 參考文獻 [1] RaffoGV,OrtegaMG,RubioFR.Anintegralpredictive/nonlinear Hcontrol structure foraquadrotorhelicopter.Automatica2010;46:29–39. [2] Derafa L,BenallegueA,FridmanL.Supertwistingcontrolalgorithmforthe attitude trackingofafourrotorsUAV.JFranklInst2012;349:685–99. [3] Luque-VegaL,Castillo-ToledoB,LoukianovAG.Robustblocksecondorder sliding modecontrolforaquadrotor.JFranklInst2012;349:719–39. [4] Garcia-Delgado L,DzulA,Santibá?ezV,LlamaM.Quadrotorsformation bansed onpotentialfunctionswithobstacleavoidance.IETControlTheory Appl 2012;6(12):1787–802. [5] AlexisK,NikolakopoulosG,TzesA.Modelpredictivequadrotorcontrol: attitude, altitudeandpositionexperimentalstudies.IETControlTheoryAppl 2012;6(12):1812–27 [6] XcvNekoukar V, Erfanian A. Adaptive fuzzy terminal sliding mode control for a class of MIMO uncertain nonlinear systems. Fuzzy Sets Syst 2011;179:34–9. [7] SAshrafiuon H, Scott-Erwin R. Sliding mode control of under-actuated multi- body systems and its application to shape change control. Int J Control 2008;81(12):1849–58. [8] Jin Y, Chang PH, Jin ML, Gweon DG. Stability guaranteed time-delay control of manipulators using nonlinear damping and terminal sliding mode. IEEE Trans Ind Electron2013;60(8):3304–17. [9] Zhi-hong M, Paplinski AP, Wu HR. A robust MIMO terminal sliding mode control scheme for rigid roboticmanipulators.IEEETransAutomControl 1994;39(12):2464–9. [10] WangL,ChaiT,ZhaiL.Neural-network-basedterminalsliding-modecontrol of roboticmanipulatorsincludingactuatordynamics.IEEETransIndElectron 2009;56(9):3296–304 [11] TanCP,YuX,ManZ.Terminalslidingmode observers foraclassofnonlinear systems.Automatica2010;46(8):1401–4. [12] Wu Y,YuX,ManZ.Terminalslidingmodecontroldesignforuncertain dynamic systems.SystControlLett1998;34:281–7. [13] Man Z,O’Day M,YuX.Arobustadaptiveterminalslidingmodecontrolfor rigid roboticmanipulators.JIntellRoboticSyst1999;24:23–41. [14] Feng Y,YuX,ManZ.Non-singularterminalslidingmodecontrolofrigid manipulators. Automatica2002;38:2159–67. [15] Hong Y,XuY,HuangJ.Finite-timecontrolforrobotmanipulators.SystControl Lett 2002;46:243–53. [16] Yu S, Yu X, Shirinzadeh B, Man Z. Continuous finite-time control for robotic manipulators with terminal sliding mode. Automatica2005;41:1957–64. [17] Chen SY, Lin FJ. Robust nonsingular terminal sliding-mode control for non- linear magnetic bearing system. IEEE Trans Control SystTechnol 2011;19(3): 636–43. [18] Chen M, Wu QX, CuiRX. Terminal sliding mode tracking control for a class of SISO uncertain nonlinear systems. ISA Trans2013;52:198–206. [19] Xu R, ?zgüner ü. Sliding mode control of aquadrotor helicopter. In:Proceed- ings of the 45th IEEE conference on decision and control, San Diego, USA.vol. 12:2006.p.4957–62. [20] Mokhtari A,BenallegueA,OrlovY.Exact linearization andslidingmode observer foraquadrotorunmannedaerialvehicle.IntJRobotAutom 2006;21:39–49. [21] Benallegue A,MokhtariA,FridmanL.High-ordersliding-modeobserverfora quadrotorUAV.IntJRobustNonlinearControl2008;18:427–40. [22] SharifiF,MirzaeiM,GordonBWZhangYM.Faulttolerantcontrolofa quadrotorUAVusingslidingmodecontrol.In:Proceedingsoftheconference on controlandfaulttolerantsystems.Nice,France;2010.p.239–44. [23] Besnard L,ShtesselYB,LandrumB.Quadrotorvehiclecontrolviaslidingmode controller drivenbyslidingmodedisturbanceobserver.JFranklInst 2012;349:658–84. [24] Efe MO.NeuralnetworkassistedcomputationallysimplePIDcontrol ofa quadrotorUAV.IEEETransIndInforma2011;7(2):354–61. [25] Lim H,ParkJ,LeeD,KimHJ.Buildyourownquadrotor:open-sourceprojects on unmannedaerialvehicles.IEEERobotAutomMag2012;19(3):33–45. [26] Zuo Z.Trajectorytrackingcontroldesignwithcommand-filtered compensation foraquadrotor.IETControlTheoryAppl2010;4(11):2343–55. [27] Das A,LewisF,SubbaraoK.Backsteppingapproachforcontrollingaquadrotor using Lagrangefromdynamics.JIntellRobotSyst2009;56:127–51. [28] AlexisK,NikolakopoulosG,TzesA.Switchingmodelpredictiveattitude control foraquadrotorhelicopter subject toatmosphericdisturbances. Control EngPract2011;19(10):1195–207. [29] Xu R,?zgünerü.Slidingmodecontrolofaclassofunderactuatedsystems. Automatica2008;44:233–41. [30] YuX,ManZ.Fastterminalsliding-modecontrol design fornonlinear dynamical systems.IEEETransCircSyst—Fundam TheoryAppl2002;49(2): 261–4. 18 Contents lists available at ScienceDirect ISA Transactions journal homepage: www.elsevier.com/locate/isatrans ISA Transactions ? (????) ???–??? Research Article Position and attitude tracking control for a quadrotor UAV Jing-Jing Xiong n, En-Hui Zheng College of Mechanical and Electrical Engineering, China Jiliang University, Hangzhou 310018, PR China Please cite this article as: Xiong J-J, Zheng E-H. Position and attitude tracking control for a quadrotor UAV. ISA Transactions (2014), http: //dx.doi.org/10.1016/j.isatra.2014.01.004i a r t i c l e i n f o Article history: Received 3 November 2013 Received in revised form 29 December 2013 Accepted 16 January 2014 This paper was recommended for publication by Jeff Pieper. Keywords: Quadrotor UAV Underactuated Novel robust TSMC SMC Synthesis control a b s t r a c t A synthesis control method is proposed to perform the position and attitude tracking control of the dynamical model of a small quadrotor unmanned aerial vehicle (UAV), where the dynamical model is underactuated, highly-coupled and nonlinear. Firstly, the dynamical model is divided into a fully actuated subsystem and an underactuated subsystem. Secondly, a controller of the fully actuated subsystem is designed through a novel robust terminal sliding mode control (TSMC) algorithm, which is utilized to guarantee all state variables converge to their desired values in short time, the convergence time is so small that the state variables are acted as time invariants in the underactuated subsystem, and, a controller of the underactuated subsystem is designed via sliding mode control (SMC), in addition, the stabilities of the subsystems are demonstrated by Lyapunov theory, respectively. Lastly, in order to demonstrate the robustness of the proposed control method, the aerodynamic forces and moments and air drag taken as external disturbances are taken into account, the obtained simulation results show that the synthesis control method has good performance in terms of position and attitude tracking when faced with external disturbances. & 2014 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction The quadrotor unmanned aerial vehicles (UAVs) are being used in several typical missions, such as search and rescue missions, surveillance, inspection, mapping, aerial cinematography and law enforcement [1–5]. Considering that the dynamical model of the quadrotor is an underactuated, highly-coupled and nonlinear system, many con- trol strategies have been developed for a class of similar systems. Among them, sliding mode control, which has drawn researchers' much attention, has been a useful and ef?cient control algorithm for handling systems with large uncertainties, time varying prop- erties, nonlinearities, and bounded external disturbances [6]. The approach is based on de?ning exponentially stable sliding surfaces as a function of tracking errors and using Lyapunov theory to guarantee all state trajectories reach these surfaces in ?nite-time, and, since these surfaces are asymptotically stable, the state trajectories slides along these surfaces till they reach the origin [7]. But, in order to obtain fast tracking error convergence, the desired poles must be chosen far from the origin on the left half of s-plane, simultaneously, this will, in turn, increase the gain of the controller, which is undesirable considering the actuator satura- tion in practical systems [8,9]. n Corresponding author. E-mail addresses: jjxiong357@gmail.com (J.-J. Xiong), ehzheng@cjlu.edu.cn (E.-H. Zheng). Replacing the conventional linear sliding surface with the non- linear terminal sliding surface, the faster tracking error convergence is to obtain through terminal sliding mode control (TSMC). Terminal sliding mode has been shown to be effective for providing faster convergence than the linear hyperplane-based sliding mode around the equilibrium point [8,10,11]. TSMC was proposed for uncertain dynamic systems with pure-feedback form in [12]. In [13], a robust adaptive TSMC technique was developed for n-link rigid robotic manipulators with uncertain dynamics. A global non-singular TSMC for rigid manipulators was presented in [14]. Finite-time control of the robot system was studied through both state feedback and dynamic output feedback control in [15]. A continuous ?nite-time control scheme for rigid robotic manipulators using a new form of terminal sliding modes was proposed in [16]. For the sake of achieving ?nite-time tracking control for the rotor position in the axial direction of a nonlinear thrust active magnetic bearing system, the robust non-singular TSMC was given in [17]. However, the conventional TSMC methods are not the best in the convergence time, the primary reason is that the convergence speed of the nonlinear sliding mode is slower than the linear sliding mode when the state variables are close to the equilibrium points. In [18], a novel TSMC scheme was developed using a function augmented sliding hyperplane for the guarantee that the tracking error converges to zero in ?nite-time, and was proposed for the uncertain single-input and single-output (SISO) nonlinear system with unknown external disturbance. In the most of existing research results, the uncertain external disturbances are not taken into account these nonlinear systems. In order to further demonstrate the robustness of novel 0019-0578/$ - see front matter & 2014 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2014.01.004 quadrotor is set up in this work by the body-frame B and the earth- frame E as presented in Fig. 1. Let the vector [x,y,z]' denotes the position of the center of the gravity of the quadrotor and the vector [u, v,w]' denotes its linear velocity in the earth-frame. The vector [p,q,r]' represents the quadrotor's angular velocity in the body-frame. m denotes the total mass. g represents the acceleration of gravity. l denotes the distance from the center of each rotor to the center of gravity. The orientation of the quadrotor is given by the rotation matrix R:E-B, where R depends on the three Euler angles [?,θ,ψ]0 , which represent the roll, the pitch and the yaw, respectively. And ?Ae π=2; π=2T; θ Ae π=2; π=2T; ψ Ae π; πT. The transformation matrix from [?,θ,ψ]0 to [p,q,r]0 is given by 2 p 3 2 1 0 sin θ 32 ?_ 3 6 q 7 6 0 cos ? sin ? cos θ 76 θ_ 7 4 5 ? 4 56 7 e1T r 0 sin ? cos ? cos θ 4 ψ_ 5 Fig. 1. Quadrotor UAV. The dynamical model of the quadrotor can be described by the following equations [5,24,29]: 8 x€ ? 1 e cos ? sin θ cos ψ t sin ? sin ψ Tu1 K1 x_ m m > TSMC, the external disturbances are considered into the nonlinear > y€ ? 1  K2 y_ 1 > me cos ? sin θ sin ψ sin ? cos ψ Tu m > systems and are applied to the controller design. > z€ ? 1  K3 z_ <> me cos ? cos θTu1 g m In this work, we combine two components in the proposed ?€ _ Iy Iz Jr θ_ Ω l K4 l?_ e2T control: a novel robust TSMC component for high accuracy > ? θψ_ > Ix t Ix r t Ixu2 Ix tracking performance in the fully actuated subsystem, and a SMC > θ€ ? ψ_ ?_ Iz Ix Jr ?_ Ωr t l u3 K5 lθ_ component that handles the external disturbances in the under- > > > ψ€ Iy Iy _ _ Ix Iy 1 Iy Iy K6 actuated subsystem. Even though many classical, higher order and extended SMC :> ? ?θ Iz t Izu4 Iz ψ_ strategies, have been developed for the ?ight controller design for where Ki denote the drag coef?cients and positive constants; Ωr ? Ω1 Ω2 t Ω3 Ω4; Ωi ; stand for the angular speed of the the quadrotor UAV (see for instance [19–23], and the list is not exhaustive), and, these strategies in the papers [19–23] were propeller i Ix,Iy,Iz represent the inertias of the quadrotor;Jr denotes the inertia of the propeller;u1 denotes the total thrust on the body utilized to dictate a necessity to compensate for the external disturbances, in addition, the other control methods, such as in the z-axis;u2 and u3 represent the roll and pitch inputs, respectively;u4 denotes a yawing moment.u1 ? eF1 t F2 t F3 t F4T; proportional–integral–differential (PID) control [24,25], backstep- ping control [26,27], switching model predictive attitude control u2 ?e F2 tF4T; u3 2 ?e F1 t F3T; u4 ? de F1 t F2 t F3 t F4T=b;, [28], etc., have been proposed for the ?ight controller design, most of the aforementioned control strategies have been proposed in order to make the quadrotor stable in ?nite-time and the stabili- zation time of the aircraft may be too long to re?ect the performance of them. In addition, the stabilization time is essen- tial signi?cance for the quadrotor UAV to quickly recover from some unexpected disturbances. For the sake of decreasing the time, a synthesis control method based on the novel robust TSMC and SMC algorithms is applied to the dynamical model of the quadrotor UAV. The synthesis control method is proposed to guarantee all system state variables converge to their desired where Fi ? bΩi denote the thrust generated by four rotors and are considered as the real control inputs to the dynamical system, b denotes the lift coef?cient;d denotes the force to moment scaling factor. 3. Synthesis control Compared with the brushless motor, the propeller is very light, we ignore the moment of inertia caused by the propeller. Eq. (2) is divided into two parts: values in short time. Furthermore, the convergence time of the state variables are predicted via the equations derived by the novel " z€ # 2 u1 cos ? cos θ 3 2 g m  K3 3 z_ T m 3 robust TSMC, this is demonstrated by the following sections. ψ€ ? 4 1 5t4 ?_ θ_ Ix Iy K6 5 e The organization of this work is arranged as follows. Section 2 presents the dynamical model of a small quadrotor UAV. The 8 " x€ # > Izu4 " cos ψ sin ψ u1 Iz Iz ψ_ #" cos ? sin θ #  x_ 2 K1 3 — m synthesis control method is detailedly introduced in Section 3. In Section 4, simulation results are performed to highlight the overall >> y€ ? m > < sin ψ cos ψ sin ? t4 K2 5 y_ — m validity and the effectiveness of the designed controllers. In > " ?€ # " l=Ix 0 #" u # 2 θ_ ψ_ Iy Iz I  K4 l _ 3 ? I e4T Section 5, a discussion, which is based on different synthesis > 2 x x control schemes, is presented to emphasize the performance of > θ€ ? 0 l=Iy u3 t4 ψ_ ?_ Iz Ix K5 lθ_ 5 >: the proposed synthesis control method in this work, followed by  Iy Iy the concluding remarks in Section 6. 2. Quadrotor model In order to describe the motion situations of the quadrotor model clearly, the position coordinate is to choose. The model of the where Eq. (3) denotes the fully actuated subsystem (FAS), Eq. (4) denotes the underactuated subsystem (UAS). For the FAS, a novel robust TSMC is used to guarantee its state variables converge to their desired values in short time, then the state variables are regarded as time invariants, therefore, the UAS gets simpli?ed. For the UAS, a sliding mode control approach is utilized. The special synthesis control scheme is introduced in the following sections. J.-J. Xiong, E.-H. Zheng / ISA Transactions ? (????) ???–??? 7 3.1. A novel robust TSMC for FAS Considering the symmetry of a rigid-body quadrotor, therefore, we get Ix ? Iy Let x1 ? ?zψ ]0 and x2 ? ? ?z_ψ_ ]0 . The fully actuated Considering the Lyapunov function candidate 2 V 1 ? s2=2 η sm1 =n1 _ Invoking Eqs. (8a) and (9a) the time derivative of V1 is derived subsystem is written by V_ 1 ? s2s_2 ? s2e ε1s2 1 2 t K3z=mT ( x_ 1 ? x2; 2 0 m1 t n1 T=n1 x_ 2 ? f 1 t g1u1 t d1 e5T ? ε1s2  η1se 2 where f 1 ?? g 0]0 ; g1 ?? cos ? cos θ=m 0; 0 1=Iz ]; u1 ? ?u1 u4]0 and d1 ?? K3z_=m K6ψ_ =Iz ]0 : To develop the tracking control, the sliding manifolds are de?ned as [18,30] s2 ? s_1 tω1s1 tξ1s1m1 =n1 e6aT Considering that (m1 tn1) is positive even integer, that’s, V_ 1 r0: The state trajectories of the subsystem converge to the desired equilibrium points in ?nite-time. Therefore, the subsystem is asymptotically stable. 3.2. A SMC approach for UAS 0 0 s4 ? s_3 tω2s3 tξ2s3m2 =n2 e6bT In this section, the details about sliding mode control of a 0 0 class of underactuated systems are found in [29]. Let where s1 ? zd z; s3 ? ψ d ψ, Zd and ψd are the desired values of state variables Z and ψ, respectively. In addition, the coef?cients " cos ψ sin ψ Q ? u1 # , and y1 ? Q ?x y] ; y2 ? Q  ?x_ y_ ] ; eω1; ω2; ξ1; ξ2T are positive, m0 ; m0 ; n0 ; n0 are positive odd integers m sin ψ cos ψ 1 0 1 0 with m0 o n0 and m0 o n0 . 1 2 1 2 y ? ?? θ]0 ; y ? ??_ θ_ ]0 . The underactuated subsystem is written in 1 1 2 2 3 4 Let s2 ? 0 and s4 ? 0. The convergence time is calculated as a cascaded form follows: 1 m1 T=n1 ! y_ 1 ? y2; ξ n0 ω1?s1e0T]en0 0 0 t y_ 2 ? f 2 td2; 1 1 ts1 ? ω n0 0 ln e7aT _ 1e 1 m1T n0 ξ1 ω2?s3e0T]en2 m2 T=n2 tξ ! y3 ? y4; y_ 4 ? f 3 tg2u2 t d3: e11T 0 0 0 2 2 According to Eqs. (9a) and (9b) we can select the appropriate ts3 ? ω ln 2 m0 2en0 2T ξ2 e7bT parameters to guarantee the control law u1 and yaw angle ψ In accordance with Eq. (5) and the time derivative of s2 and s4, we have converge to the desired/reference values in short time. That’s,u_ 1 ? 0,ψ becomes time invariant, then ψ_ ? 0, Q is time invariant matrix and non-singular because u1 is the total thrust u1 K3 d m0 =n0 s_2 ? z€d cos ? cos θ t g t z_ tω1s_1 tξ1 s 1 1 e8aT m m dt 1 and nonzero to overcome the gravity. As a result 2 2 3 2 f ?? cos ? sin θ sin ?]0 ; d2 ? Q 1diag?K1=m K2=m]Qy ; f ? 0; g 1 K6 d m0 =n0 2 2 s_4 ? ψ€ d I u4 t I ψ_ tω2s_3 tξ2dts e8bT ? diag?l=Ix l=Iy]; u ? ?u u ]0 ; d ? diag? lK =I lK =I ]y 3 2 2 3 3 z z 4 x 5 y 4 The controllers are designed by De?ne the tracking error equations m m0 em0 n0 T=n0 m =n 8 e1 ? yd y1; cos ? cos θ 1 1 1 1 u1 ? z€d t g tω1s_1 tξ1  1s 1 n0 1 s_1 tε1s2 tη1s2 e9aT > 1 > < 1 1 > e2 ? e_ 1 ? y_ d d y2; e12T > ／ m0 m0 n0 T=n0 m2 =n2 ＼ e3 ? e_ 2 ? y€ 1 f 2; u4 ? Iz ψ€ d tω2s_3 tξ2 2se 2 2 2 s_3 tε2s4 tη2s e9bT > n0 3 4 > :::d ’ ?f 2  ?f 2  ?f 2 2 > e4 ? e_ 3 ? y1 ?y y2 t?y f 2 t?y y4 : where ε1,ε2,η1, and η2 are positive,m1, n1, m2, and n2 are positive 1 2 3 d odd integers with m1 on1 and m2 on2: Under the controllers, the state trajectories reach the areas (Δ1,Δ2) of the sliding surfaces s2 ? 0 and s4 ? 0 along s_2 ? ε1s2 where the vector y1 denotes the desired value vector. The sliding manifolds are designed as s ? c1e1 tc2e2 tc3e3 te4 e13T η0 m1 =n1 0 m2 =n2 1s2 and s_4 ? ε2s4 η2s4 in ?nite-time, respectively. The time is de?ned as where the constants ci 40. By making s_ ? MsgnesT λs, we get n1 ε1?s1e0T]en1 m1 T=n1 tδ1 ::: h i t0 ln e10aT 8 c1e2 tc2e3 tc3e4 t y0 ?f 2 y 9 1 rε n m δ d d < > 1 y 2 > 1e 1 1T 1 u2 ?  ?f 2 g ?y 2  1>>  d h ?f 2 i f dt ?y2 2 d h?f 2 i y dt ?y3 4 dt ? 1 > > = e14T 3 t0 2 > > n2 ε2?s3e0T]en2 m2 T=n2 tδ2 > ?f > 2 rε 2en2 ln m2T δ2 e10bT >: ?y3 ef 3 td3Tt MsgnesTtλs >; where η0 m1 =n1 m1 =n1 where M ? ec2d2 tc3β d2TjjE1jj2 tβ d4jjξeyTjj2 tρ; 1 ? η1 te K3z_=mT=js2 j; η1 ? L1=js2 jtδ1; 2 3 L1 ? jK3z_=mjmax; δ1 40; Δ1 ? fjs2jreL1=η1T m1 =n1 g β1 Z?f 2=?y1; β2 Z?f 2=?y2; β3 Z?f 2=?y3; η0 m2 =n2 m2 =n2 E1 ? ?e1e2e3] ; ξeyT? ?y1y2y3y4] and λ 40; 2 ? η2 te K6ψ_ =Iz T=js4 j; η2 ? L2=js4 jtδ2 ρ 0 d 0 o d E 0 d max K mK m m2 =n2 4 ; jj 2jj 2jj 1jj2; 2 ? e 1= 2= T L2 ? jK6ψ_ =Iz jmax; δ2 40; Δ2 ? fjs4jreL2=η2T g jjd3jjo d4jjξeyTjj2; d4 ? maxelK4=IxlK5=IyT: Proof 1. In order to illustrate the subsystem is stable, here, we According to?f 2 ? ［ sin ? sin θ cos ? cos θ cos ?0]; only choose the state variable z as an example and Lyapunov and 0 o ｜｜?f 2=?y3｜｜ ? ｜ cos 2? cos θo2, and , therefore, ?f 2= theory is applied. ｜｜ ?y3 ?y3 is invertible. ｜｜ ｜ reference real 2 x ( m ) 0 -2 0 5 10 15 20 25 30 35 40 45 50 1 y ( m ) 0 13 X: 39.54 Y: 9.801 12 u ( m/s 2 ) 11 10 1 9 8 7 0 5 10 15 20 25 30 35 40 45 50 Time ( s ) Fig. 4. The controller u1, PID control and SMC. -1 0 5 10 15 20 25 30 35 40 45 50 6  Table 1 Quadrotor model parameters. z ( m ) Variable Value Units m 2.0 kg Ix ? Iy 1.25 Ns2/rad Iz 2.2 Ns2/rad K1 ? K2 ? K3 0.01 Ns/m K ? K ? K 0.012 Ns/m l 0.20 m Jr 1 Ns2/rad b 2 Ns2 d 5 N ms2 g 9.8 m/s2 3 0 0 5 10 15 20 25 30 35 40 45 50 Time ( s ) Fig. 2. The positions (x,y,z), PID control and SMC.  4 5 6 reference real 0.025 f ( rad ) 0 -0.025 .06 Variable Value Variable Value ω1 ξ1 1 1 ω2 ξ2 3 1 m0 5 0 5 0.06 n1 0 7 n2 7 5 10 15 20 25 30 35 40 45 50 m1 1 m2 1 1 n1 3 n2 3 ε1 10 ε2 10 X: 25.48 Y: 0.5003 η1 L = sm1 =n1 η2 L2=j m2 =n2 0.5 δ1 0.1 δ2 0.1 c1 20 c2 22 c3 8 ρ 1 0 λ 0.1 β1 0 5 10 15 20 25 30 Time ( s ) 35 40 45 50 β2 0 β3 2 0  0 5 10 15 20 25 30 35 40 45 50 Table 2 Controller parameters. q ( rad ) 0 1 m2 0 - 0 y ( rad ) 1 j 2 jtδ1 s4 jtδ2 0 Fig. 3. The angles (?,θ,ψ), PID control and SMC. Proof 2. The stability of the subsystem is illustrated by Lyapunov theory as follows. Consider the Lyapunov function candidate: V 1sT s ? 2 Invoking Eqs. (13) and (14), the time derivative of V is V_ ? sT s_ ? sT ?c1e_ 1 tc2e_ 2 tc3e_ 3 t e_ 4] 4. Simulation results and analysis In this section, the dynamical model of the quadrotor UAV in Eq. (2) is used to test the validity and ef?ciency of the proposed synthesis control scheme when faced with external disturbances. The simulations of typical position and attitude tracking are performed on Matlab 7.1.0.246/Simulink, which is equipped with a computer comprising of a DUO E7200 2.53 GHz CPU with 2 GB of RAM and a 100 GB solid state disk drive. Moreover, the perfor- mance of the synthesis control is demonstrated through the comparison with the control method in [29], which used a rate ／ ? sT MsgnesTtc2d2 tc3  ?f 2 d 2 ?y 2 t  ?f 2 d ＼ 3 ?y3 b